Influence of excited state decay and dephasing on phonon quantum state preparation

The coupling between single-photon emitters and phonons opens many possibilities to store and transmit quantum properties. In this paper, we apply the independent boson model to describe the coupling between an optically driven two-level system and a discrete phonon mode. Tailored optical driving allows not only to generate coherent phonon states, but also to generate coherent superpositions in the form of Schrödinger cat states in the phonon system. We analyze the influence of decay and dephasing of the two-level system on these phonon preparation protocols. We find that the decay transforms the coherent phonon state into a circular distribution in phase space. Although the dephasing between two exciting laser pulses leads to a reduction of the interference ability in the phonon system, the decay conserves it during the transition into the ground state. This allows to store the phonon quantum state properties in the ground state of the single-photon emitter.

Figure 1

Schematics of the system and its temporal evolution. (a) Mutual influence of the generating functions caused by a laser pulse with phase $\varphi$ [Eq. (A1)] and during the evolution between the pulses [Eq. (8)]. The labels at the arrows indicate the phase that is transferred in the respective coupling. (b) Illustration of the phonon potentials associated with the ground state $|g\rangle$ (blue) and the excited state $|x\rangle$ (red) of the TLS. The phonon state is depicted by the green dots. Laser excitation and decay induce transitions between the electronic states.

Figure 2

Partition of the phonon Wigner function into the (a) excited state and (b) ground state subspace. The corresponding subspace Wigner functions are shown at five different times after excitation by a $\pi$ pulse. The system parameters are $\gamma =2,\mathrm{\Gamma }=0.5{\omega }_{\mathrm{ph}}$.

Figure 3

Phonon Wigner function at four different times after excitation by a single $\pi$ pulse for an electron-phonon coupling $\gamma =2$ and decay constants (a) $\mathrm{\Gamma }=0.5{\omega }_{\mathrm{ph}}$ and (b) $\mathrm{\Gamma }=2{\omega }_{\mathrm{ph}}$.

Figure 4

(a) Phonon Wigner function after excitation by a single $\pi$ pulse at the time $t/t_{\mathrm{ph}}=10$, i.e., after the excited state occupation has completely decayed for an electron-phonon coupling $\gamma =2$ and five different values of the decay constant $\mathrm{\Gamma }$. (b) Expectation value of the displacement $\langle u\rangle$ as a function of time after the laser pulse for the same values of $\mathrm{\Gamma }$. The dashed line shows the expectation value for the case without decay (i.e., $\mathrm{\Gamma }=0$).

Figure 5

Phonon state after a two-pulse excitation. The pulse areas are ${\theta }_i=\pi /2$, the delay is $t_2-t_1=t_{\mathrm{ph}}/2$, and the depicted time is $t/t_{\mathrm{ph}}=3/4$ after the second pulse. (a) Wigner function for the coupling strengths (a) $\gamma =2$ and (b) $\gamma =0.5$. The pure dephasing rate increases from $\tilde{\beta }=0$ (left) via $0.1{\omega }_{\mathrm{ph}}$ (center) to $0.5{\omega }_{\mathrm{ph}}$ (right).

Figure 6

EIFs $S_U$ (blue) and $S_{\mathrm{\Pi }}$ (red) as functions of time after the second pulse. The phonon coupling strength is $\gamma =0.5$, and the pure dephasing rate increases from dark to bright lines. The dashed line is the limiting case of a statistical mixture, i.e., $\tilde{\beta }\gg {\omega }_{\mathrm{ph}}$.

Figure 7

Excited state decay induced decay of a cat state. Snapshots of the Wigner function's dynamics for a cat state prepared in the excited state $|x\rangle$. The decay rate is $\mathrm{\Gamma }=0.2{\omega }_{\mathrm{ph}}$, and the coupling strength is $\gamma =2$.

Figure 8

Phonon Wigner function resulting from the excited state-induced decay of a cat state at the time $t/t_{\mathrm{ph}}=9.5$, i.e., after the excited state occupation has completely decayed for an electron-phonon coupling $\gamma =2$ and five different values of the decay constant $\mathrm{\Gamma }$.

Figure 9

Schematic of the Wigner function in the ground state subspace at times $t/t_{\mathrm{ph}}=n+\frac{1}{2}(n\ge 1)$. The four horizontally aligned cat states are examples for cat states resulting from decay processes at different times. The numbers indicate the time of the decay (in units of $t/t_{\mathrm{ph}}$). For a continuous decay, the interference patterns are continuously distributed along the dotted circle.

Figure 10

Cat state generation with excited state decay and dephasing. Snapshots of the Wigner function's dynamics during the first phonon period after the second laser pulse ($t=0\to t=t_{\mathrm{ph}}$) and at the time $t/t_{\mathrm{ph}}=9/2$ (bottom right). The decay rate is $\mathrm{\Gamma }=0.2{\omega }_{\mathrm{ph}}$, and the coupling strength is $\gamma =2$.