Qubits as Probes for Topological Materials

A study of decoherence

Nicolas Delnour1, Richard MacKenzie1, Michael Hilke2, Hichem Eleuch3
Université de Montréal1, McGill University2, University of Sharjah3

Introducing Concepts

Topological insulator (TI) TIs can be defined as materials acting as insulators in the bulk while hosting symmetry protected topological (SPT) states at their boundaries.
Fig. 1 - SSH chain for N=20 [2].
Fig. 2 - Comparison of a bulk vs edge solutions to the Schrödinger equation for an SSH chain of length N=21.
Fig. 3 - Energy spectrum as a function of hopping parameter ratio r for N=21. Solid (dashed) lines are the bulk (edge) energies. The shaded grey region shows energy bands in the thermodynamic limit [3].

A qubit or double dot (DD) is a two-level system (TLS) in a coherent superposition of two states. The Hamiltonian for a qubit is expressed simply as \[ H^{D D}=\left(\begin{array}{cc} \epsilon_{1} & \tau \\ \tau & \epsilon_{2}\end{array}\right) \] with eigenenergies λ ±. The transition probability between both levels can be found from the off-diagonal element of the Green's function G12 with respect to time: \[ G^{\mathrm{DD}}(E)=(E-H^{\mathrm{DD}})^{-1} \] \[ G_{12}^{\mathrm{DD}}(t)=\int_{-\infty}^{\infty} d E e^{-i E t} G_{12}^{\mathrm{DD}}(E) \]

Tripartite System

We consider an open system shown in Figure 4 consisting of a qubit coupled (coupling strength tc) to an SSH chain to which is, in turn, coupled a semi-infinite lead (coupling strength tL).

Fig. 4 - Tripartite system including a qubit, SSH chain, and semi-infinite conducting lead. The qubit is coupled at an arbitrary nth site.

The Hamiltonian for such a system has the form: \[ H=\left(\begin{array}{c|c|c} H^{DD} & V_N & 0 \\ \hline V_N^{\dagger} & H_{\mathrm{SSH}} & W \\ \hline 0 & W^{\dagger} & H_{\infty} \end{array}\right) \]

As we are interested in the behaviour of the qubit, we can incorporate the impact of the environment into a self-energy term [4] such that: \[ H^{D D \prime}=\left(\begin{array}{cc} \epsilon_{1} & \tau \\ \tau & \epsilon_{2}+\Sigma_{\mathrm{SSH},\infty}^{nn}(E) \end{array}\right) \] where \[ \Sigma_{\mathrm{SSH},\infty}^{nn}(E) = t_{c}^{2} G_{\mathrm{SSH},\infty}^{nn}(E) \]

\[ \left(\tau_{\phi}\right)^{-1} \approx \min \left(-\frac{1}{2} \Im\left\{\Sigma^{nn}_{\mathrm{SSH}, \infty}\left(\lambda_{\pm}\right) \pm \delta^{\prime}\left(\lambda_{\pm}\right)\right\}\right) \]

Proposed Measurement

This sweep through the SSH chain allows us to characterise the decoherence as a function of site number for a given qubit tuning.

Fig. 5 - Decoherence rate of a qubit coupled to the SSH chain (length N = 69 with single left edge state) and lead system. Note two different regimes: 1) oscillatory behaviour associated to coupling with bulk states from λ+= 0.4044. 2) exponential decay from coupling of edge state with λ-= 0.
Information gained

\[ e^{\alpha_{decay} n} \sim |\psi_{edge} |^2 \]

Discussion and future avenues for research

Where to look from here? Why care? This analysis considers a "toy-system" in order to evaluate decoherence of an open system within reasonable analytical approximations, but qubit behaviours should extend to more complex systems. Qubits could therefore be useful for: