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Abstract
Topological protection is a promising way to deal with the decoherence issue in quantum systems. To induce topological excitations in Josephson junction arrays (JJA), we propose a scheme where only Josephson critical currents are tuned in JJA on honeycomb lattice, which is easily achieved by changing potential barriers or lengths of insulators between superconducting islands. In the present system, a hexagonal cluster with six superconducting islands forms a unit cell, and the difference between intra- and inter-hexagon Josephson critical currents yields a topologically nontrivial state accompanied by a band inversion. Topological Josephson plasmon modes appear at the interface between topological and trivial domains, carrying pseudospin dominant energy flows, which mimics the spin-momentum locking in the renowned quantum spin Hall effect.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Recently, topological insulators and their variants attract increasing interests [1–3]. One of the characteristic features of topological systems is the appearance of edge states, protected/guaranteed by the topology in bulk band structures. For instance, in a quantum spin Hall insulator [4] (2D topological insulator), one observes robust edge states with their propagation directions fixed by spin. Namely, the up- and down-spin edge modes propagate in opposite directions. This property can potentially lead to spintronics applications [1,2]. While it was first developed in electronic systems, the notion of topology also promotes fruitful works in photonic systems [5–10].
Characterized by a phase of superconducting long-range order parameter, superconductors are macroscopic quantum materials [11–13]. In previous proposals [14–16], topological protection is discussed as a possible route to suppress decoherence in Josephson junction systems, which may be useful for quantum information processing. A Josephson junction is referred as two superconducting islands (or, superconducting nodes) separated by an insulator or a potential barrier. The superconducting current between the two islands is proportional to $\sin (\phi _j-\phi _i)$ (dc Josephson relation), and with a finite voltage $V$ across the junction the phase difference evolves with time as $d(\phi _j-\phi _i)/dt \propto 2eV/\hbar$ (ac Josephson relation) [17–21]. When a phase difference appears in a Josephson junction, it generates currents between nodes, and causes charging of the nodes. Then, the charging energy of the nodes with effective capacitance acts as a restoring force, and leads to oscillations in charge of the superconductors around their equilibrium values. In an array of Josephson junctions, there appear Josephson plasmon modes.
In this paper, we demonstrate a simple scheme to realize topological plasmon modes in a Josephson junction array (JJA) on honeycomb lattice, which only requires tuning of Josephson critical currents. As demonstrated in Ref. [22], Dirac cones are formed at the $K$ and $K'$ points of the Brillouin zone (BZ) in the granular superconductors on honeycomb lattice. We introduce real-space patterns in Josephson critical currents that preserve the $C_{6v}$-symmetric texture [6], which opens a frequency band gap at the Dirac cones. From mode symmetry around the band gap, it is found that a band inversion takes place when the inter-hexagon Josephson critical currents are larger than the intra-hexagon ones, indicating a nontrivial topology. Topological Josephson plasmon modes appear at the interface between topological and trivial domains of JJA, where the direction of time-averaged energy flow is determined by the orbital angular momentum defined on the hexagonal unit cell [6,23,24], mimicking the spin-momentum locking in the renowned quantum spin Hall effect.
2. Josephson plasmon modes described by charge dynamics
A Josephson junction is constructed by two superconductors $i$ and $j$ separated by an insulator or a potential barrier. According to the dc Josephson relation [17–20], supercurrent flows through the Josephson junction even in the absence of voltage difference between superconductors,
where $\phi _i$ and $\phi _j$ are superconducting phases of superconductors $i$ and $j$ respectively, and $I_{\textrm{c}ij}$ is the Josephson critical current between them. When a voltage difference $V_i-V_j$ is applied, the phase difference $\phi _j-\phi _i$ changes with time due to the ac Josephson relation [17–20] where $\hbar$ is the reduced Planck constant and $e$ is the charge of electron.We consider a JJA with superconductors shunted individually to a common ground by capacitors with a common capacitance $C$ [see Fig. 1]. Tuning the Fermi level at zero energy inside the superconducting gap, there is no charge at superconducting nodes at equilibrium. The charge accumulated on site $i$ at time $t$ is given by integrations of currents
where $j$ runs over neighboring sites connected to site $i$. The voltage measured from the ground at site $i$ is related to the accumulated charge $q_i$ viaThe equation of motion for voltage is derived by combining Eqs. (1)–(4) as
with $\omega ^2_{ij}=2eI_{\textrm{c}ij}/\hbar C$, where we focus on the situation of small phase differences by adapting the approximation $\cos (\phi _j-\phi _i)\sim 1$.
Fig. 1. (a) Josephson junction array (JJA) on honeycomb lattice. Superconductors are connected to each other by Josephson junctions and are grouped into hexagonal unit cells indicated by the dashed line. Each superconductor is shunted to a common ground by a capacitor with the common capacitance $C$. The Josephson critical current is $I_{\textrm{c}0}$/$I_{\textrm{c}1}$ inside/between the unit cells. The numbers 1,…,6 specify the sites in a unit cell, and lattice vectors are $\boldsymbol {a_1}$ and $\boldsymbol {a_2}$. (b) and (c) Schematics of Josephson junctions in (a), where superconducting parts are represented by blue, and insulating parts are presented by red/green for $I_{\textrm{c}0}$/$I_{\textrm {c}1}$.
Now, we move onto the specific arrangement of the JJA for realizing topological Josephson plasmon modes. As in Fig. 1(a), we consider JJA with patterned Josephson critical currents. The modulation is such that a hexagon-shaped cluster forms a unit cell, and Josephson critical currents are assigned for junctions between ($I_{\textrm {c}1}$) and inside ($I_{\textrm {c}0}$) unit cells. Introducing $I_\textrm{c}$ as $I_\textrm{c}=(2I_{\textrm {c}0}+I_{\textrm {c}1})/3$, we can define a characteristic frequency $\omega _0=\sqrt {2eI_{\textrm{c}}/\hbar C}$. As illustrated in Figs. 1(b) and 1(c), Josephson critical currents can be tuned by changing the length of insulator or the potential barrier between superconducting nodes. In this periodically modulated JJA, a Josephson plasmon mode as oscillations of voltages with oscillation frequency $\omega$ is represented in the form $V=[V_1, V_2, V_3, V_4, V_5, V_6]^T\textrm {e}^{\textrm{i}(\boldsymbol {k}\cdot \boldsymbol {r}-\omega t)}$, where 1-6 label the sites inside a unit cell [see Fig. 1(a)] and $\boldsymbol {r}$ is the position of the unit cell. The secular equation is obtained from Eq. (5) as
where the $6\times 6$ matrix $H_{\boldsymbol {k}}$ is given by3. Topological Josephson plasmon modes
By solving Eqs. (6)–(8), we can derive frequency band structures of Josephson plasmon modes for JJA, as shown in Figs. 2(a) and 2(b). For $I_{\textrm {c}0}=I_{\textrm {c}1}$, the JJA resembles graphene [22,25], where Dirac cones are formed at the $K$ and $K'$ points in the original BZ for the pristine honeycomb lattice. Using the enlarged unit cell for $I_{\textrm {c}0}\neq I_{\textrm {c}1}$ [see Fig. 1(a)], Dirac cones at the $K$ and $K'$ points in the original BZ are folded on to the $\Gamma$ point in the new BZ, leading to the four-fold degeneracy as can be seen in the frequency band structure denoted by dashed lines in Figs. 2(a) and 2(b). Because the modulated honeycomb lattice considered here preserves the $C_{6v}$ symmetry, eigenstates at the $\Gamma$ point can be described by the two two-dimensional irrecucible representations of the $C_{6v}$ point group, namely $p$ and $d$ modes defined on the hexagonal unit cell as illustrated in Figs. 2(c)–2(f). The four-fold degeneracy for $I_{\textrm {c}0}=I_{\textrm {c}1}$ is lifted by tuning Josephson critical currents into $I_{\textrm {c}0}$ and $I_{\textrm {c}1}$, resulting in two double degeneracies. One can define a pseudo time-reversal symmetry from the real time-reversal symmetry and the $C_{6v}$ symmetry, which leads to the doublets $p_{\pm }=\frac {1}{\sqrt {2}}(p_x\pm \textrm{i}p_{y})$ and $d_{\pm }=\frac {1}{\sqrt {2}}(d_{x^2-y^2}\pm \textrm{i}d_{2xy})$ [6]. The orbital angular momenta associated with these eigen wavefunctions play the role of electron spins in the quantum spin Hall effect as will be revealed below.
Fig. 2. (a) Frequency band structure of Josephson plasmon modes for $I_{\textrm {c}0}>I_{\textrm {c}1}$, where a gap opens with $d$ bands above the gap and $p$ bands below the gap with $\omega _0=\sqrt {2eI_{\textrm{c}}/\hbar C}$. The wavefunction parity of the inversion symmetry is denoted by $+$ (even) and $-$ (odd). (b) Same as (a) except for $I_{\textrm {c}0}<I_{\textrm {c}1}$, where a band inversion takes place at the $\Gamma$ point. Dashed lines in (a) and (b) denote the gapless frequency band structure of Josephson plasmon modes for the artificial graphene ($I_{\textrm {c}0}=I_{\textrm {c}1}$). We chose parameters satisfying that $2I_{\textrm {c}0}+I_{\textrm {c}1}=3I_\textrm{c}$ and $|I_{\textrm {c}0}-I_{\textrm {c}1}|=0.3I_\textrm{c}$, to make the band edges in (a) and (b) aligned. (c)-(f) Wavefunctions of $p$ and $d$ waves at the $\Gamma$ point, with red/blue color for positive/negative component. (g) and (h) Schematic pictures of time-averaged energy flow of Josephson plasmon mode with up- and down-pseudospin, respectively, which are defined as $p_{\pm }=\frac {1}{\sqrt {2}}(p_x\pm \textrm{i}p_{y})$ and $d_{\pm }=\frac {1}{\sqrt {2}}(d_{x^2-y^2}\pm \textrm{i}d_{2xy})$.
The state with $I_{\textrm {c}0}>I_{\textrm {c}1}$ is topologically distinct from the one with $I_{\textrm {c}0}<I_{\textrm {c}1}$, due to the $p$-$d$ band inversion [6]. One possible way to capture the topology in this system is to count the parity of eigenstate at the $\Gamma$ and $M$ points, where the spatial inversion symmetry is preserved [26]. For $I_{\textrm {c}0}>I_{\textrm {c}1}$, the even-/odd-parity $d$/$p$ modes appear at the upper/lower band edge, as shown in Fig. 2(a). For each band, the parity at the $M$ point is the same as that at the $\Gamma$ point, signaling a topologically trivial state. On the other hand, for $I_{\textrm {c}0}<I_{\textrm {c}1}$, the even-/odd-parity $d$/$p$ modes appear at the lower/upper band edge, as shown in Fig. 2(b). This causes imbalance of the number of modes with even and odd parity, namely, below the band gap, there are three eigenstates with even parity at the $\Gamma$ point while there is only one eigenstate with even parity at the $M$ point. The unequal numbers of eigenstates with even parity feature a topological state that can be transformed from the trivial one only via a band gap closing.
Fig. 3. (a) JJA including an interface between a trivial domain and a topological domain. The Josephson critical currents within/between unit cells in the trivial domain are denoted by red/light brown lines, whereas in the topological domain by green/dark brown lines. At the interface, Josephson critical currents are the geometric mean of values of $I_{\textrm {c}1}$ in Figs. 2(a) and 2(b). A supercell is chosen as shown in the rectangular frame with $\boldsymbol {a}_\textrm{s}$ as the unit vector. (b) Frequency band structure of Josephson plasmon modes calculated based on the supercell given in (a), where topological interface dispersions (red and blue lines) appear in the bulk band gap.
Governed by the energy conservation in the present system, decrease of the energy on site $i$ is given as
where $S_{ij}$ is the energy flowing from site $i$ to site $j$. By using Eq. (3) and Eq. (4), one has $S_{ij}=V_iI_{ij}$. Note that $V_i$ and $I_{ij}$ are actually real parts of those obtained from Eq. (6). From Eq. (1) and Eq. (2), it can be seen that $I_{ij}$ is the real part of $\textrm{i}2eI_{\textrm{c}ij}(V_i-V_j)/\omega \hbar$ for a Josephson plasmon mode with frequency $\omega$. Therefore $S_{ij}$ becomesThe time-averaged energy flow is then obtained as
Using this relation, it is shown that the time-averaged energy flows of the eigenstates $p_{\pm }$ and $d_{\pm }$ circulate counterclockwise and clockwise in unit cells respectively, as shown in Figs. 2(g) and 2(h).
Now we reveal the topological Josephson plasmon modes at the interface between a trivial domain and a topological domain. As illustrated in Fig. 3(a), the considered system is uniform and infinitely long in the $x$ direction, allowing us to calculate the band structure along the $x$ direction. In the $y$ direction, trivial and topological domains each containing 60 unit cells are arranged periodically. We adapt the parameters in Figs. 2(a) and 2(b) respectively for the topological and trivial domains satisfying $2I_{\textrm {c}0}+I_{\textrm {c}1}=3I_\textrm{c}$ and $|I_{\textrm {c}0}-I_{\textrm {c}1}|=0.3I_\textrm{c}$, where the bulk band gaps perfectly overlap. Topological interface dispersions obtained by using the supercell denoted in the rectangular frame in Fig. 3(a) appear inside the bulk band gap as shown in Fig. 3(b).
To examine the wavefunctions and time-averaged energy flows associated with the interface modes, we pick up two in-gap modes with the same frequency and the opposite momenta. These two modes have the same magnitude of wavefunction, which decay into the bulks exponentially as shown in Figs. 4(a) and 4(b). The time-averaged energy flows are represented as arrows between sites in Fig. 4, showing opposite circulations in unit cells. In parallel with Figs. 2(g) and 2(h), the counterclockwise and clockwise circulations in unit cells respectively correspond to up- and down-pseudospins. As illustrated in Figs. 4(a) and 4(b), the net energy flow for the up/down-pseudospin points to the $+/-x$ direction, manifesting the pseudospin-momentum locking effect in these topological Josephson plasmon modes.
Fig. 4. Distributions of wavefunction amplitude (size of dot) and time-averaged energy flow (arrow between sites) for the topological Josephson plasmon mode at the momentum denoted by $+$ (a) and $-$ (b) in Fig. 3(b).
4. Quantum version of topological Josephson plasmon modes
So far, we have studied the Josephson plasmon modes as oscillations of charge using a circuit representation. In this section, we briefly discuss on its quantum version.
The Hamiltonian of a JJA is
where the first summation runs over all nearest neighbor sites, $J_{ij}$ stands for the Josephson couplings and $U$ is the uniform charging energy of superconductors with $n_i$ being the Cooper pair number operator. We focus on the regime with $J_{ij}\gg U$, where phase shifts $\phi _i$ from the global coherent phase of the ground state are small.Neglecting the irrelevant constant and higher order terms of phase shifts, we obtain
Using the same setup as shown in Fig. 1(a), one can replace $I_{\textrm {c}0}$ and $I_{\textrm {c}1}$ by Josephson couplings $J_0$ and $J_1$ respectively, and the charging energies are represented as the term proportional to $U$. Similar to discussions in Sec. 2, we take the hexagonal unit cells and apply the Fourier transformation to obtain the Hamiltonian in momentum space as
where $\Phi _{\boldsymbol {k}}=[\phi _{1,{\boldsymbol {k}}},\phi _{2,{\boldsymbol {k}}},\phi _{3,{\boldsymbol {k}}},\phi _{4,{\boldsymbol {k}}},\phi _{5,{\boldsymbol {k}}},\phi _{6,{\boldsymbol {k}}}]$ and $N_{\boldsymbol {k}}=[n_{1,{\boldsymbol {k}}},n_{2,{\boldsymbol {k}}},n_{3,{\boldsymbol {k}}},n_{4,{\boldsymbol {k}}},n_{5,{\boldsymbol {k}}},n_{6,{\boldsymbol {k}}}]$, $I_6$ is a $6 \times 6$ identity matrix,Starting from the Heisenberg equations,
5. Discussion
In the present study, we only consider JJAs with small charging energy $U$. It is important to point out that JJAs go through a superconductor-insulator phase transition at a threshold value of $U$ and become insulating. In that case, phase fluctuations will destroy the global coherent phase and Cooper pairs are basically bound at individual superconducting nodes, where plasmon modes are hardly excited. That is why we focus on the regime $J_{ij}\gg U$.
JJAs proposed in the present work can be experimentally fabricated by conventional photolithgraphy or evaporation techniques [27]. Particularly, experimental realization of JJA with square and triangular lattices have been reported in Ref. [28]. The same techniques can be extended to honeycomb lattice with textures in critical current required in our present work. The homogeneous Josephson junctions in Ref. [28] are made by two overlapping superconducting layers separated by a thin oxide layer. Textures in critical current can be achieved by tuning the thickness of oxide layer between superconducting layers.
On the other hand, the supercurrent $I_{ij}$ and voltage $V_i$ can be measured locally with method similar to those reported in Ref. [29]. One can thus determine energy flows between nodes and check the pseudospin-momentum locking in topological JJA as predicted theoretically in our present work.
6. Conclusion
In summary, we propose a method to achieve topological plasmon modes in Josephson junction arrays on honeycomb structure. For uniform Josephson critical current, dispersions of Josephson plasmon modes are similar to those of graphene. Making the Josephson critical currents within a hexagonal unit cell smaller/larger than those between unit cells, we can obtain the topological/trivial band gap associated with a $p$-$d$ band inversion. Topological Josephson plasmon modes appear at the interface between trivial and topological domains, decaying into the bulks exponentially. The propagation direction of the net energy flow of a topological Josephson plasmon mode is governed by orbital angular momentum defined on the hexagonal unit cell, manifesting the pseudospin-momentum locking effect.
Disclosures
The authors declare no conflicts of interest.
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