Qubits as Edge State Detectors: Illustration Using the SSH Model
635
Lead
SSH chain
TLS
t L
t 1
t 2
t C
τ
t 1
t 2
ϵ 1
ϵ 2
Fig. 1 Tripartite system geometry. Rightmost SSH chain hopping parameter is t 1 or t 2 depending
on whether number of sites N is even or odd, respectively (odd case shown here)
states. As we will see, the presence of edge states greatly increases the decoherence
rate.
2 Two-Level System: A Rapid Review
We review the isolated TLS, mostly to establish notation to be used in what follows.
The TLS Hamiltonian is
H
DD
=
2 τ
τ τ 1
≡
0 − δ 0 /2
τ
τ
τ 0 + δ/2
.
(1)
The energies are λ ± =
1
2 (( 1 + 2 ± δ) = 0 ±
δ
2 , where δ =
(( 1 − 2 ) 2 + 4τ 2 .
The energy-dependent Green’s function is defined by G DD (E) = (E − H DD ) −1 ;
its Fourier transform gives the time-dependent Green’s function, which is a sum of
oscillatory terms with frequencies given by the energies; for instance,
G
DD
12 (t) = −
2πiτ
δ
e
−iλ + t
− e
−iλ − t
.
(2)
When we couple the TLS to the rest of the system, it will decohere; this will be seen
in the Green’s function, which will exhibit decaying behavior [1].
3 Su–Schrieffer–Heeger Model and Edge States
The SSH Hamiltonian [2], proposed in the context of the polymer polyacetylene for
reasons we will not go into here, is