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

Abstract

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.

Entropy Dynamics of Phonon Quantum States Generated by Optical Excitation of a Two-Level System

“…Therefore, for the phonons, we can only calculate linear entropies, according to Equation ( 20 ). In Reference [ 32 ], we explained how the Wigner function evolves during the decay process in the TLS. Therefore, we consider the same optical excitation with a pulse area of , which initially fully inverts the TLS.…”

“…The phonon quantum state gets more involved when a two-pulse excitation is considered. As extensively studied in Reference [ 13 , 32 ], an excitation with two pulses having pulse areas of and a delay of leads to the generation of two Schrödinger cat states, each in one TLS subspace. A Schrödinger cat state is a coherent superposition of two coherent states, i.e., of the most classical states of a harmonic oscillator, and, as such, it is of high interest in all areas of quantum optics [ 39 , 40 , 41 ] and, more recently, also phononics [ 11 ].…”

“…For reasons of clarity, we consider a single Schrödinger cat state entirely in the excited state as initial state without any optical excitation and account for a decay of the TLS into its ground state. Although this state cannot directly be prepared by optical pulses, in Reference [ 32 ] it is explained how it is constructed mathematically as initial state for the simulated decay dynamics. Some snapshots of the corresponding Wigner function dynamics are shown in Figure 7 a for a small decay rate of .…”

“…In the same way as in Reference [ 32 ], we model a decay of the excited state with the rate via the Lindblad dissipator leading to the master equation for the density matrix: …”

“…As we have explained in Reference [ 32 ], the entire quantum state and especially the Wigner function of the phonons can be calculated analytically when considering a series of ultrafast laser pulses to drive the TLS. Especially if the pulse duration is much shorter than the phonon period, the pulses can be approximated by delta-functions as …”