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## exciton photon matrix element

Here, we are using second quantization. Please refer to this paper http://prb.aps.org/abstract/PRB/v75/i3/e035405
Hamiltonian of electron-photon is
$H_{el-op}=\sum_k D_k c^+_{kc}c_{kv}(a+a^+)$

$c^+_{kc}c_{kv}$ are creation of electron to conduction band and annihilation electron in valence band, respectively. $(a+a^+)$ are photon annihilation and creation operator.
Exciton wave function is
$|\Psi^f\rangle=\sum_k Z^n_{k_{c},k_v}c^+_{k_{c}}c_{k_v}|0\rangle$
Where Z is weighting coefficient, kc is electron state (conduction band), kv is hole state (valence band), and $|0\rangle$ is ground state (all electrons occupy valence) band.

Matrix element of exciton-photon is defined as
$M_{ex-op}=\langle\Psi^f|H_{el-op}|0\rangle$

$M_{ex-op}=\sum_k Z^{n*}_{k_{c},k_v}D_k\langle 0|a+a^+|0\rangle$
My question is, how can we prove that $\langle 0|a+a^+|0\rangle=1$ to get

$M_{ex-op}=\sum_k Z^{n*}_{k_{c},k_v}D_k$
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