The excitonic qubit coupled with a phonon bath on a star graph: anomalous decoherence and coherence revivals

Abstract

Based on the operatorial formulation of perturbation theory, the dynamical properties of a Frenkel exciton coupled with a thermal phonon bath on a star graph are studied. Within this method, the dynamics is governed by an effective Hamiltonian which accounts for exciton–phonon entanglement. The exciton is dressed by a virtual phonon cloud, whereas the phonons are dressed by virtual excitonic transitions. Special attention is paid to the description of the coherence of a qubit state initially located on the central node of the graph. Within the nonadiabatic weak coupling limit, it is shown that several timescales govern the coherence dynamics. In the short time limit, the coherence behaves as if the exciton was insensitive to the phonon bath. Then, quantum decoherence takes place, this decoherence being enhanced by the size of the graph and by temperature. However, the coherence does not vanish in the long time limit. Instead, it exhibits incomplete revivals that occur periodically at specific revival times and it shows almost exact recurrences that take place at particular super-revival times, a singular behavior that has been corroborated by performing exact quantum calculations.

Appendices

Appendix 1: Quasi-degenerate second-order perturbation theory

As detailed previously [54], the generator of the unitary transformation up to second order in V is given by the following equations

$$\begin{aligned} \left[ H_0,S_1 \right]= & {} V_{NC}, \nonumber \\ \left[ H_0,S_2 \right]= & {} \frac{1}{2} [ S_1,V ]_{NC}, \nonumber \\ \hat{H}= & {} H_0+\frac{1}{2}[ S_1,V]_{C}, \end{aligned}$$

(27)

where the symbol C means phonon-conserving part of the operator, whereas the symbol NC means phonon-nonconserving part of the operator. From the expression of the exciton–phonon interaction Hamiltonian V, one easily obtains

$$\begin{aligned} S_1= & {} \sum _{\ell =0}^N Z^{(\ell )} a_\ell ^{\dag }-Z^{(\ell )\dag } a_\ell , \nonumber \\ S_2= & {} \sum _{\ell \ell '} E^{(\ell \ell ')}a_\ell ^{\dag }a_{\ell '}^{\dag }-E^{(\ell \ell ')\dag }a_{\ell '}a_{\ell } . \end{aligned}$$

(28)

In the exciton eigenbasis, the various operators that enter the definition of the generator are defined in terms of the coupling operator $M^{(\ell )}=\varDelta _{0}|\ell \rangle \langle \ell |$ as

$$\begin{aligned}&[H_\mathrm{A},Z^{(\ell )}]+\varOmega _0 Z^{(\ell )} = M^{(\ell )}, \nonumber \\&A^{(\ell \ell ')}= [Z^{(\ell )},M^{(\ell ')}]/2 , \nonumber \\&[H_\mathrm{A}, E^{(\ell \ell ')}] + 2\varOmega _0 E^{(\ell \ell ')} = A^{(\ell \ell ')} . \end{aligned}$$

(29)

Appendix 2: Excitonic coherences

In the excitonic eigenbasis $\{|\chi _i\rangle \}$, the approximate expression for the effective exciton propagator up to second order in the exciton–phonon coupling is written as

$$\begin{aligned} G_{ij}(t)= & {} \delta _{ij} \frac{\mathcal {Z}_\mathrm{B}^{(i)}(t)}{\mathcal {Z}_\mathrm{B}} \ \mathrm{e}^{-i\hat{\epsilon }_{i}t} - \frac{\mathcal {Z}_\mathrm{B}^{(i)}(t)}{2\mathcal {Z}_\mathrm{B}}\ \mathrm{e}^{-i\hat{\epsilon }_{i}t} \left[ \sum _{\ell ,\ell '}\sum _{i'}\left( Z^{(\ell )}_{ii'}Z_{i'j}^{(\ell ')\dagger } +Z_{ii'}^{(\ell ')\dagger } Z^{(\ell )}_{i'j} \right) \right. \nonumber \\&\times \sum _{q} \beta _{q,i}^*(\ell ) \beta _{q,i}(\ell ')\ n_{q,i}(t) \left. + \sum _\ell \sum _{i'} Z_{ii'}^{(\ell )\dagger } Z^{(\ell )}_{i'j} \right] \nonumber \\&- \frac{\mathcal {Z}_\mathrm{B}^{(j)}(t)}{2\mathcal {Z}_\mathrm{B}}\ \mathrm{e}^{-i\hat{\epsilon }_{j}t} \left[ \sum _{\ell ,\ell '}\sum _{i'}\left( Z^{(\ell )}_{ii'}Z_{i'j}^{(\ell ')\dagger } +Z_{ii'}^{(\ell ')\dagger } Z^{(\ell )}_{i'j} \right) \right. \nonumber \\&\times \sum _{q} \beta _{q,j}^*(\ell ) \beta _{q,j}(\ell ')\ n_{q,j}(t) \left. + \sum _\ell \sum _{i'} Z_{ii'}^{(\ell )\dagger } Z^{(\ell )}_{i'j} \right] \nonumber \\&+ \sum _{i'}\frac{\mathcal {Z}_\mathrm{B}^{(i')}(t)}{\mathcal {Z}_\mathrm{B}} \mathrm{e}^{-i\hat{\epsilon }_{i}t} \sum _{\ell ,\ell '} \left[ Z^{(\ell )}_{ii'}Z_{i'j}^{(\ell ')\dagger } \sum _{q} n_{q,i'}(t)\ \mathrm{e}^{i(\varOmega _0+\delta \varOmega _{q,i'} ) t} \beta _{q,i'}^*(\ell ) \beta _{q,i'}(\ell ') \right. \nonumber \\&\left. + Z_{ii'}^{(\ell ')\dagger } Z^{(\ell )}_{i'j} \sum _{q} (n_{q,i'}(t)+1) \ \mathrm{e}^{-i (\varOmega _0+\delta \varOmega _{q,i'})t} \beta _{q,i'}(l) \beta _{q,i'}^*(l') \right] , \end{aligned}$$

(30)

where $n_{q,i}(t) = (\mathrm{e}^{\beta \varOmega _0+i\varOmega _{q,i}t} - 1)^{-1}$. In Eq. (30), $Z_{ij}^{(\ell )}$ and $Z_{ij}^{(\ell )\dag }$ are the matrix elements in the excitonic eigenbasis of the operator $Z^{(\ell )}$ and $Z^{(\ell )\dag }$ defined in “Appendix 1.” At this step, the effective propagator $G_{\ell \ell _0}(t)$ is obtained by performing a change of basis.