Detecting topological edge states with the dynamics of a qubit
Introduction
Qubits are the building blocks of any quantum information processing device. Two of the most challenging problems for quantum computing and other applications are decoherence due to the interaction with environment and perturbations due to manufacturing imperfections [1], [2], [3]. These effects limit the effective performance of quantum devices, such as the speed of an eventual quantum computer. Thus, evaluating the decoherence rate for the qubit or for an ensemble of coupled qubits is of great importance.
In previous work [4], the decay rate of a qubit coupled to another system with or without disorder was studied. The main objective was to investigate under which circumstances the interaction of a qubit with its surroundings can be designed to improve the qubit's performance in a quantum device by increasing the decoherence time. It was shown that the decoherence rate of the qubit is related to transport properties of the coupled system. Furthermore, it was proven that disorder lowers the decoherence rate on average. This suggests potential applications to increase the performance of qubits in a quantum device by adding impurities to the system.
In this work, a similar composite system is studied from a different perspective. Rather than viewing the qubit as the system of interest and tailoring the system with which it interacts to improve the qubit's decoherence, the qubit is viewed as a measuring device capable of determining properties of its environment. In particular, we explore the dynamics of a qubit attached to a Su-Schrieffer-Heeger (SSH) chain which is then attached to a third system modeling the environment; the third system is a standard tight-binding hopping Hamiltonian (without dimerization).
The SSH model, described in detail below, is one of the simplest systems exhibiting interesting topology such as solitons and, of interest here, edge states [5], [6], [7], [8], [9], [10]. In spite of the model's inherent simplicity, it manages to capture many interesting and important physical effects in topological systems. The model has also been extended to study topological insulators of higher dimensions [11]. Normally, almost-zero-energy edge states have exponentially localized wavefunctions at the edges. These states are a particular type of topological edge states. Topological edge states have captured the interest of researchers in several fields of physics due to their diverse surprising proprieties. To name but a few examples, they can enhance the sound intensity at phononic crystal interfaces [12], allow a robust one-way propagation [13] or protect light transport in nanophotonics systems [14]. For further applications and references, see [15], [16].
We evaluate the decoherence rate of the qubit and how it depends on the properties of the SSH system in order to probe the edge states of the SSH chain. As we will see, it is strongly affected by such states at the qubit end of the coupled system.
In the next section, we review the isolated SSH model, mainly to establish notation but also to highlight the conditions for the existence of edge states and their properties. In Section 3, we explore the double dot coupled to an SSH chain which is itself coupled to semi-infinite chain. An expression for the decoherence rate is derived using a semi-analytic approximation which is in excellent agreement with numerical simulations of the same system. We will see a strong effect of edge states on the decoherence rate. We conclude with a discussion of our results and avenues for future work in Section 4.
Section snippets
Su-Schrieffer-Heeger model
The SSH model [17] is a one-dimensional tight-binding model with alternating coupling strengths due to the Peierls instability [18], [19], leading to a parity effect in the chain length. The Hamiltonian for a chain of N sites is where or for N even or odd, respectively. We will assume for simplicity.
SSH chain coupled to semi-infinite lead and qubit
In this section we will consider the full system consisting of the SSH chain coupled on the left to a semi-infinite lead and on the right to a qubit (Fig. 5). Since we will ultimately incorporate the effects of the SSH chain and semi-infinite lead into an effective qubit Hamiltonian, in the next subsection we will quickly review the relevant features of the qubit. We will then study the full system in the following subsection.
Discussion and conclusions
Interestingly, the exponential decrease in the rate of decoherence is similar for the case where we have an edge state on the opposite side of the qubit (N odd in Fig. 8) and when there are no edge states at all (N even in Fig. 7). In both cases, the decoherence rate is exponentially suppressed. This is quite similar to the case of localization with a random potential along the chain, which also leads to an exponential suppression of the decoherence rate with chain length [4]. From this
CRediT authorship contribution statement
Meri Zaimi: Conceptualization, Formal analysis. Christian Boudreault: Conceptualization, Formal analysis. Nouédyn Baspin: Software. Nicolas Delnour: Software, Writing – review & editing. Hichem Eleuch: Conceptualization, Formal analysis, Methodology. Richard MacKenzie: Conceptualization, Formal analysis, Funding acquisition, Methodology, Supervision, Writing – original draft, Writing – review & editing. Michael Hilke: Conceptualization, Formal analysis, Funding acquisition, Methodology,
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was supported in part by the Natural Science and Engineering Research Council of Canada and by the Fonds de Recherche Nature et Technologies du Québec via the INTRIQ strategic cluster grant. RM is grateful for the hospitality of Perimeter Institute where part of this work was carried out. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of
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